How to Straighten Curved UVs into Straight Belt
Select all wanted faces and Unwrap them.
Select one face on the mesh.
Align this face in UV Image Editor by selecting edges and W > Align Auto.
Select all faces on your mesh (keep aligned face as active face).
Unwrap it by Fallow Active Quads.


Edit:

Differences between this answer and answer from Duarte Farrajota Ramos.

As for number of steps it could tempting to believe that fewer is better, but not always. Here is object unwrapped with pure Follow Active Quad > Pack Islands and my method. Not additional steps was taken.

Duarte Farrajota Ramos answer:



This answer: